inordinatum

Physics and Mathematics of Disordered Systems

Estimating expected growth

Let’s take some fluctuating time series — say the value of some financial asset, like a stock price. What is its average growth rate? This seemingly trivial question came up in a recent discussion I had with a friend; obviously, it is relevant for quantitative finance and many other applications. Looking at it in more detail, it turns out that a precise answer is actually not that simple, and depends on the time range for which one would like to estimate the expected growth. So I’d like to share here some insights on this problem and its interesting connections to stochastic processes. Suprisingly, this was only studied in the literature quite recently!

The setup

Let’s consider the price $s_t$ of an asset at $N+1$ discrete times $t=1...N+1$. The corresponding growth rates $r_t$ are defined by

$\displaystyle 1+r_t := \frac{s_{t+1}}{s_t}.$    (1)

I.e. if $s_{t+1}=1.5 s_t$, the growth rate is $r_t=0.5=50\%$. Let’s also assume that our growth process $r_t$ is stationary, i.e. that the distribution of the growth rates does not change in time. Say for concreteness that the growth rates $r_t$ are i.i.d. (independent identically distributed) random variables.

The arithmetic average

The most immediate idea for computing the average growth rate $\overline{r}$ is just to take the arithmetic average, i.e.

$\displaystyle \overline{r}^{AM} := \frac{1}{N}\sum_{t=1}^N r_t.$    (2)

What does this give us? Obviously, by the assumption of stationarity made above, with increasing sample size $N$ the expectation value for the growth rate at the next time step $r_{N+1}$ (or any other future time step) is approximated better and better by $\overline{r}^{AM}$:

$\displaystyle \lim_{N\to\infty} E[r_{N+1}]-\frac{1}{N}\sum_{t=1}^N r_t =0.$

This seems to be what we’d expect for an average growth rate, so what’s missing? Let’s go back to our sample of $N$ asset prices and take the total growth

$\displaystyle \frac{s_{N+1}}{s_1} = \prod_{i=1}^N (1+r_i).$

By the relationship between the arithmetic and the geometric mean, we know that

$\displaystyle \frac{s_{N+1}}{s_1} = \prod_{i=1}^N (1+r_i) \leq \left[\frac{1}{N}\sum_{i=1}^N (1+r_i)\right]^N = \left(1+\overline{r}^{AM}\right)^N.$ (3)

The left-hand side is the total growth after $N$ time periods and the right-hand side is its estimate from the arithmetic mean in eq. (2). Equality only holds in eq. (3) when all the $r_t$ are equal. When the $r_t$ fluctuate the total growth rate will always be strictly less than the estimate from the arithmetic mean, even as $N \to \infty$. So, by taking the arithmetic mean to obtain the total growth in the price of our asset at the end of the observation period, we systematically overestimate it. The cause for this is the compounding performed to obtain the total growth over more than one time step. This makes our observable a nonlinear function of the growth rate in a single time step. Thus, fluctuations in the growth rate don’t average out, and yield a net shift in its expectation value.

The first explicit observation of this effect I’ve found is in a 1974 paper by M. E. Blume, Unbiased Estimators of Long-Run Expected Rates of Return. Quantitative estimates from that paper and from a later study by Jacquier, Kane and Marcus show that in typical situations this overestimation may easily be as large as 25%-100%.

The geometric average

So we see that the arithmetic mean does not provide an unbiased estimate of the compounded growth over a longer time period. Another natural way to obtain the average growth, which intuitively seems more adapted to capture that effect, is to take the geometric average of the growth rates:

$\displaystyle 1+\overline{r}^{GM} := \left(\prod_{t=1}^N 1+r_t\right)^{1/N}.$    (4)

Now, by construction, the total growth at the end of our observation period, i.e. after $N$ time periods, is correctly captured:

$\displaystyle \frac{s_{N+1}}{s_1} = \left(1+\overline{r}^{GM}\right)^N$

But this only solves the problem which we observed after eq. (3) for this specific case, when estimating growth for a time period whose length is exactly the same as the observation time span $N$ from which we obtain the average. For shorter time periods (in particular, when estimating the growth rate for a single time step $r_t$), the geometric mean will now underestimate the growth rate. On the other hand, for time periods even longer than the observation period, it will still overestimate it like the arithmetic mean (see again the paper by Blume for a more detailed discussion).

Unbiased estimators

Considering the results above, the issue becomes clearer: A reasonable (i.e. unbiased) estimate for the compounded growth over a time period requires a formula that takes into account both the number of observed time steps $N$, and the number of time steps over which we’d like to estimate the compounded growth $T$. For $T=1$ we can use the arithmetic mean, for $T=N$ we can use the geometric mean, and for general $T$ we need another estimator altogether. For the general case, Blume proposes the following (approximately) unbiased estimator $\left(\overline{r}^{UB}\right)^T$:

$\displaystyle \overline{r}^{UB} = \frac{N-T}{N-1}\left(1+\overline{r}^{AM}\right)^T+\frac{T-1}{N-1}\left(1+\overline{r}^{GM}\right)^T.$    (5)

Eq. (5) is a reasonable approximation for the compounded growth, not having any further information on the form of the distribution, correlations, etc.
For $T=1$, i.e. estimating growth in a single time step, this gives just the arithmetic mean $\overline{r}^{AM}$ which is fine as we saw above. For $T=N$, this gives the geometric mean which is also correct. For other values of $T$, eq. (5) is a linear combination of the arithmetic and the geometric mean.

To see how the coefficients in eq. (5) arise, let us start with an ansatz of the form

$\displaystyle \left(\overline{r}^{UB}\right)^T = \alpha\left(1+\overline{r}^{AM}\right)^T+\beta\left(1+\overline{r}^{GM}\right)^T.$    (6)

Let us further split up the growth rates as $r_t = \mu + \epsilon_t$, where $\mu$ is the “true” average growth rate and $\epsilon$ are fluctuations. Inserting this as well as the definitions of $\overline{r}^{AM}, \overline{r}^{GM}$ into eq. (6) we get

$\displaystyle \left(\overline{r}^{UB}\right)^T = \alpha\left(1+\mu +\frac{1}{N}\sum_{i=1}^N\epsilon_i\right)^T+\beta\left[\prod_{i=1}^N (1+\mu+\epsilon_i)\right]^{T/N}.$

Now let us assume that the fluctuations $\epsilon$ are small, and satisfy $E[\epsilon_i\epsilon_j]=\sigma^2 \delta_{ij}$. Expanding to second order in $\epsilon$ (the first order vanishes), and taking the expectation value, we obtain

$\begin{array}{rl} \displaystyle E\left[\left(\overline{r}^{UB}\right)^T\right] = & \displaystyle \alpha\left(1+\mu\right)^T\left[1+\frac{T(T-1)\sigma^2}{2(1+\mu)^2N}\right] \\ & \displaystyle +\beta\left(1+\mu\right)^T\left[1+\frac{T(T-N)\sigma^2}{2(1+\mu)^2N}\right] \\ & \displaystyle + \mathcal{O}(\sigma^4). \end{array}$

To obtain an estimator that is unbiased (to second order in $\sigma$), we now choose $\alpha$ and $\beta$ such that the term of order $\sigma^0$ is just the true growth rate $(1+\mu)^T$, and the term of order $\sigma^2$ vanishes. This gives the system

$\displaystyle \begin{array}{rl} \displaystyle \alpha + \beta & = 1,\\ \displaystyle \frac{T-1}{2}\alpha+\frac{T-N}{2}\beta & = 0. \end{array}$

The solution of this linear system for $\alpha$ and $\beta$ yields exactly the coefficients in eq. (5).
Of course, here we make the assumption of small fluctuations and also a specific ansatz for the estimator. If one has more information on the distribution of the growth rates $r$ this may not be the most adequate one, but with what we know there’s not much more we can do!

Outlook

As you can see from the above discussion, estimating the expected (compounded) growth over a series of time steps is more complex than it appears at first sight. I’ve shown some basic results, but didn’t touch on many other important aspects:

• In addition to the growth rate $r$, it is also interesting to consider the discount factor $r^{-T}$. Blume’s approach is extended to this observable in this paper by Ian Cooper.
• If one assumes the growth factors $1+r$ in eq. (1) to be log-normally distributed, the problem can be treated analytically. Jacquier, Kane and Marcus discuss this case in detail in this paper, and also provide an exact result for the unbiased estimator.
• The assumption of independent, identically distributed growth rates is not very realistic. On the one hand, we expect the distribution from which the annual returns are drawn to vary in time (i.e. due to underlying macroeconomic conditions). This is discussed briefly in Cooper’s paper. On the other hand, we also expect some amount of correlation between subsequent time steps, even if the underlying distribution does not change. It would be interesting to see how this modifies the results above.

Let me know if you find these interesting — I’ll be glad to expand on that in a future post!